C₇×D₅×C₁₃⋊D₁₃ #37556884

order	23660
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂×C₁₃

Aliases

direct product	C₇×D₅×C₁₃⋊D₁₃
unambiguous semidirect product	(D₅×C₁₃)⋊(C₇×D₁₃)

Splittings

Fitting, socle, Hall	C₅×C₇×C₁₃² ⋊ C₂²
derived, Hall	C₅×C₁₃² ⋊ C₂²×C₇
Hall	C₇×C₁₃² ⋊ C₂×D₅
Sylow	C₁₃² ⋊ C₂×C₇×D₅
Hall	C₅×C₇ ⋊ C₂×C₁₃⋊D₁₃
Sylow	C₅ ⋊ C₂×C₇×C₁₃⋊D₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₅×C₁₃⋊D₁₃
commutator, cosocle, hypocenter	C₅×C₁₃²	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₃²	C₂²
LCS	G ▷ C₅×C₁₃²
UCS	C₁ ◁ C₇

320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 5915

A minimal faithful permutation representation — 3 generators on 38 points:

(1,2,4,5,3)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33,34,35,36,37,38);(2,3)(4,5)(19,20,22,24,26,28,30,31,29,27,25,23,21);(6,7,9,11,13,15,17,18,16,14,12,10,8)

Conjugacy classes & character table

2408 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23660, 114);

C7×D5×C13⋊D13 #37556884